 # Bulk Modulus of Gases

The Bulk modulus of a substance describes its resistance to uniform compression. It is the ratio of infinite increase in pressure to the resulting decrease in the volume. The SI unit of bulk modulus is pascal and dimensional form is M1L-1T-2.

In other words, the numerical constant shows the elastic properties of a fluid or solid when it is in some kind of pressure on every surface. Bulk modulus measures the ability of an object to withstand any changes in volume when compressed on every side. It equals to the quotient of pressure applied divided by the relative deformation.

The relative formation is called as the strain which is the volume changed by the original volume. Hence, if original volume V0 is reduced by the pressure applied to a new volume Vn. The strain can be expressed as change in volume V0 – Vn divided by the original volume.

## Bulk Modulus Formula

The formula to calculate bulk modulus is expressed in the following manner.

Bulk Modulus = $\frac{Pressure}{strain}=\frac{p}{(V_{o}-V_{n})/V_{o}}$

### Bulk Modulus for Gases

All gases have an equation that relates density, pressure and temperature, i.e, the equation of a state. This helps us to deduce an explicit relationship between density and pressure. Such relationships are based on the process that is considered i.e, isentropic at constant entropy or isothermal at the constant temperature. The equations deduced are expressed in the following manner.

$\frac{p}{ho}=$  constant for isothermal process                                        (1)

And

$\frac{p}{ho ^{k}}$  constant for isentropic process                                  (2)

Where k denotes the ratio of specific heat at constant pressure cp to the specific heat at constant volume cv. Moreover, cp – cv = R where R indicates gas constant. Under normal conditions, k = 1.4 for air.

When you substitute equations 1 and 2 in the bulk modulus definition, we get,

Ev = p for isothermal process

Ev = kp for isentropic process

Hence you can notice that bulk modulus of a gas is based on its pressure. The atmospheric pressure of air at STP is 1.01325 x 105N/m2 and the bulk modulus is of similar order, while it is 2.15 x 109 N/m2. These values show that air is over 15,000 times more compressible than water.